Unit vector

A vector of magnitude exactly $1$ , used to isolate direction alone.

A unit vector is any vector of magnitude $1$ . Dividing a nonzero vector by its own magnitude, $\hat{\mathbf{v}} = \mathbf{v}/|\mathbf{v}|$ , always produces one, since that division is scalar multiplication by the positive constant $1/|\mathbf{v}|$ — it strips away "how far" while leaving "which way" untouched.

Examples

For $\mathbf{v}=\langle 3,4\rangle$ , $|\mathbf{v}|=5$ , so $\hat{\mathbf{v}}=\langle 0.6,\ 0.8\rangle$ .
For $\mathbf{u}=\langle 4,0,3\rangle$ in space, $|\mathbf{u}|=5$ , so $\hat{\mathbf{u}}=\langle 0.8,0,0.6\rangle$ .